Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}7x-3y &= -5 \\ 8x-3y &= -1\end{align*}$
We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-1$ and the bottom equation by $1$ $\begin{align*}-7x+3y &= 5\\ 8x-3y &= -1\end{align*}$ Add the top and bottom equations. $x = 4$ $x = 4$ Substitute $4$ for $x$ in the top equation. $7( 4)-3y = -5$ $28-3y = -5$ $-3y = -33$ $y = 11$ The solution is $\enspace x = 4, \enspace y = 11$.